2007 IEEE International Conference on Signal Processing and Communications (ICSPC 2007), 24-27 November 2007, Dubai, United Arab Emirates
High-Order Gm-C Filters with Current Transfer Function
Based on Multiple Loop Feedback
Mohamed O.Shaker
Soliman A. Mahmoud
Ahmed M. Soliman
Electronics and Communications
Department,
Cairo University
Giza, Egypt
Electrical and Electronics Engineering
Department,
German University in Cairo
Cairo, Egypt
Electronics and Communications
Department,
Cairo University
Giza, Egypt
Abstract
Despite the wealth of literature on Gm-C filters, the
synthesis of high-order filter characteristics is still an active
topic. In this paper the realization of Gm-C filters with
current transfer functions based on multiple loop feedback
structures are presented. The new method presents 32
structures in case of third-order filter. All the filter
architectures contain minimum number of components and
all internal nodes in canonical realizations have grounded
capacitors. PSpice simulation result for band-pass filter
with current transfer function derived from the proposed
model verifying the analytical results using 0.35µm
technology is also given.
Index Terms—Gm-C filters, OTA, BOTA, DOTA,
synthesis.
I. INTRODUCTION
The performance of low-order Gm-C filters has been
proved and a body of literature has been published for the
design of second order Gm-C filters [1], [2]. Recently, the
design of high-order Gm-C filters has received a great
interest and has been thoroughly investigated by several
methods. The first method is the cascade of biquadratic
sections, which can realize any arbitrary transmission zeros
but leads to an unacceptable high sensitivity to component
parameter variations [3], [4]. The second method is the
ladder simulation; it has a very low sensitivity to
component parameter variations but can realize
transmission zeros only on the imaginary axis [5]-[7]. The
third method depends on the multiple loop feedback. This
method has both advantages, It has a low sensitivity to
component variations and can realize any arbitrary
transmission zeros [8]-[10]. Practical consideration in high
frequency Gm-C filter design may specify using grounded
capacitors and reducing the number of components [5]. The
former is because the grounded capacitor can be
implemented on a smaller area than the floating counterpart
and it can absorb equivalent shunt capacitive parasitics.
The latter is due to the fact that a large number of
components may increase power consumption, chip areas,
noise, and parasitic effects. Thus the design method and
resulting filter structures should be based on grounded
1-4244-1236-6/07/$25.00 © 2007 IEEE
capacitors and canonical architectures. The aim of this
paper is to present new high-order Gm-C filter models with
current transfer function employing only fully differential
transconductors and grounded capacitors. The first model is
capable of providing lowpass function. The second model
is obtained by inserting multiple input current sources to
the first one. Any nth-order current transfer function can be
realized by the second one, which simultaneously enjoys
several attractive criteria: the minimum components, it
needs only n fully differential transconductors to realize nth
order filter, and only 2n grounded capacitors. The symbol
of the fully differential transconductor is shown in Fig. 1 in
which the differential output current Io2-Io1 is linearly
proportional to the differential input voltage V1-V2 such
that
Io2 - Io1= G(V1 - V2)
(1)
Where G is the equivalent transconductance. In section II, the first
model and the possible generated structures are introduced.
In section III, the second model, which can realize any type
of filters, is presented. Design example is given in section
IV. Finally, conclusions are stated in section V.
V1
Io2
+
G
V2
+
-
Io1
Fig. 1: The symbol of the fully differential transconductor.
If+
-
+
If-
Ii+
+
C
C
-
Ii-
Io+
+
G
-
+
-
+
-
Io-
Fig. 2: The symbol of the differential current integrator.
85
[ ]
feedback current vector, and F = fij n*n , the feedback
II. THE LOW-PASS FILTER MODEL
The basic building block in the construction of Gm-C
filters is the integrator in which the differential output
current can
be obtained as:
G
(I i − I f )
sC
(2)
-
+
-
-
+
-
-
s
+
= C j /G j .
(6)
1
−1
Cn
Cn
j
where
Gn
+
+
where the time constant
-
Ifn
(5)
= I oj − I fj+1
j+1I oj+1
I o = M(s) −1 (BIi − I f )
+
Ion
s
s
2
−1
M(s) =
s
3
(7)
.
+
-
+
If3
C3
C3
+
- -
-
+
G2
-
+
(8)
So, the equations for the whole system can be obtained as
+
Io2
Where the system coefficient matrix is
C2
C2
(9)
A(s)I o = BI i
-
+
If2
0 ........0]t
-
+
-
+
+
Ii
-
+
C1
C1
-
If1
Fig. 3: The proposed multiple loop feedback filter model
with current transfer function.
where I o = I o+ - I o- , the differential output current,
I i = I i + - I i- , the differential input current, and I f = I f + - I f ,the differential feedback current.
The proposed nth-order multiple loop feedback model is
shown in Fig. 3. This model is composed of a feedforward
network consisting of n Gm-C integrators connected in
cascade and a feedback network that contains only pure
wire connections to keep the condition of minimum
number of components. This model can be described as
(3)
I f = FI o
(10)
A(s) = M(s) + F
G1
-
+
Io1
+
- -
n
-
And B = [1
G3
-
+
Io3
+
-
.
−1 s
+
Feedback Network
(4)
s 1I o1 = I i − I f1
Eqn. (4) and (5) can be condensed in a matrix form as
Ion=Iout
Io =
coefficient matrix.
The equations of the feedforward network can be written as
where I o = [I o1 I o2 .......I on ] , the differential output current
t
vector of integrators, I f = [I f1 I f2 .......I fn ]t , the differential
Eqn. (9) establishes the relationship between the overall
circuit input and the integrator outputs including the overall
circuit output. The circuit transfer function can be
formulated as
Io
= A(s) −1 B
Ii
A11 (s)
=
1
A(s)
A12 (s)
.
.
(11)
A1n (s)
where
A(s)
and A ij (s) represent the determinant and
cofactors of A(s) , respectively.
Note the structural feature of M(s) and that F is an upper
triangular matrix, the transfer function T(s) can be
simplified as
86
T(s) =
I out I on A1n (s)
1
=
=
=
Ii
Ii
A(s)
A(s)
(12)
The feedback matrix F is defined by eqn. (3) and has the
property that
f ij
= 1 If there is a direct connection between I fi and I oj
(13)
= 0 Otherwise
The minimum component realization, or canonical
realization, means that for realizing the nth-order all pole
lowpass filter, only n fully differential transconductors and
2n grounded capacitors are required. For the model shown
in Fig. 3, the canonical realization is clearly equivalent to
no components existing in the feedback network.
Alternatively, it can be said that for canonical architectures,
the feedback matrix F defined by eqn. (3) obviously has
only zero and unit elements, since feedback can only be
achieved by direct connections.
Eqn. (12) can realize nth order low-pass function if at least
one of the following conditions is achieved.
A. The first condition
The first condition is that feedback signal is exist in each
integrator input nodes. This condition means that the
feedback matrix F is an upper triangular matrix, which has
at least one unit element in each row.
So, the number of filter structures (m), which equals the
number of the possible feedback matrices, could be
obtained as
n
m=
∏m ,
i
i =1
m i+1 = 2m i + 1
i = 1,2,.....
model generates more filter structures, that’s because of
using multiple output transconductors and the output
current of any integrator could be feedback to many circuit
nodes. In the model given in [9], the output current of any
integrator is feedback to only one circuit node because of
using balanced output transconductors. This means that the
feedback matrix model is an upper triangular matrix, which
has one and only one unit element in each column.
Finally, the proposed model shown in Fig. 3 is a general
model which can realize nth order low-pass function if at
least one of the previous conditions is achieved and the
total number of the filters structures (m) can be obtained as
n
∏m
m=2
while the number of those derived from the model given in
[8] equals n!. It’s clear that the proposed model generates
more filter structures, that’s because any feedback current
could be the summation of two or more output currents
using only pure wires. In the model given in [8], the
feedback voltage is just one of the output voltages to keep
the condition of the minimum number of components. This
means that the feedback matrix is an upper triangular
matrix, which has one and only one unit element in each
row.
B. The second condition
The second condition is that the output signal of any
integrator is feedback to some circuit node. This condition
means that the feedback matrix F is an upper triangular
matrix, which has at least one unit element in each column.
So, the number of filter structures could be obtained as
given in eqn. (14) while the number of those derived from
the model given in [9] equals n! It’s clear that the proposed
− The number of redundant structures (15)
For the third-order model that is a derivative version of the
proposed model corresponding to n=3, with general F and
using eqn. (12), the general transfer function can be
formulated as
T(s) = 1/
{
1 2 3s
3
+(
1 2 f 33
+
+[ 1 (f 22 f 33 + f 23 ) +
1 3f 22
+
2 f11f 33
+
2 3 f11 )s
2
3 (f11f 22
+ (f11f 22 f 33 + f11f 23 + f12 f 33 + f13 )
}
+ f12 )]s
(16)
It can be found that the proposed model generates 32 filters. It’s
worth to mention that 9 structures are not practical because
they have no solution for the Butterworth approximations,
which has been numerically verified, and therefore are
rejected.
III. THE GENERAL FILTER MODEL
(14)
m1 = 1
i
i =1
The general nth-order filter model can be derived from the
proposed one by inserting different current signals to all
integrators input nodes. The exactly same formulation
process as that in section II can be followed to derive the
design equations for this model. The only one exception is
that instead of BI i = [I i 0 ........0]t , it can now be obtained
as
BI i = [I1 I 2 ........I n ]t
(17)
This exception is clearly due to the change of input form.
In Fig. 3, the input current is applied onto only the first
integrator input nodes, while in the present case the input
currents are applied onto all integrators input nodes and the
last integrator output nodes without any need to extra
transconductors.
So, the integrators output currents can be obtained as
I o = A(s) −1 [I1 I 2 ........I n ]t
and the overall output current is
87
(18)
I out = I n +1 + I on = I n +1 +
1
A(s)
n
(19)
A jn (s)I j
j=1
Note that A ij (s) is at least one order less in s than A(s) .
So, I n +1 is used to provide the nth order numerator of the
transfer function. It’s clear that the filter given in [10] is
just one case, which can be obtained from the proposed
model by selecting the feedback matrix as
0 0......... 0 1
0 0......... 0 1
(20)
F= .
.
Fig. 5: The magnitude frequency response of the thirdorder band-pass filter.
0 0......... 0 1
IV. DESIGN EXAMPLES
REFERENCES
The designed filter is a third-order band-pass filter shown
in Fig. 4 with 20 MHz center frequency. The circuit
capacitances are C1 = 0.6 pF , C 2 = 0.15 pF , C 3 = 1.2 pF and
the equal transconductance is adopted with the
transconductance value being 50 A/V . The fully
differential transconductor used for the design was given in
[11]. The magnitude frequency response is shown in Fig. 5,
which is close to the ideal characteristics.
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C1
C1
+
+
C2
-
C2
G1
-
+
+
C3
-
C3
G2
-
+
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+-
G3 +-
+
+
-
+
-
-
+
Ii
Fig. 4: The third-order band-pass Gm-C filter derived from the
model.
V. CONCLUSION
This paper has presented two new Gm-C filter models with
current transfer function. The first one provides low-pass
function and the second one generates any type of filters.
These models generate large number of filters, which keep
the condition of minimum number of components and
grounded capacitors. The key point is using current instead
of voltage to represent signal and using multiple output
transconductors. The analytical results have been
confirmed using PSpice simulations.
Iout
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